Vibrational Mpemba relaxation in a linear damped elastic system

We show that a linear damped elastic system can display a Mpemba-like reversal of vibrational relaxation without any nonlinear constitutive response or amplitude-dependent damping. The mechanism is purely modal. For a one-dimensional Kelvin–Voigt elastic medium, the decay rate of the $n$-th vibrational mode scales as ${\gamma }_n\propto n^2$. We formulate relaxation in terms of a positive modal envelope energy, thereby excluding crossings caused merely by oscillation phase. A one-parameter family of initial states is then constructed such that increasing the initial elastic excitation simultaneously reduces the projection onto the slow fundamental mode. In the invariant two-mode subspace, the relaxation-order crossing is obtained analytically and, for this family, occurs at a universal dimensionless time independent of the selected pair of initial states. In the limiting case where the slowest mode is absent, the dominant relaxation rate changes discontinuously, yielding a direct vibrational analogue of the strong Mpemba effect. We further show that the same relaxation-order reversal occurs in the ordinary mechanical vibrational energy, which decreases monotonically despite retaining the oscillatory dynamics of the underlying modes.